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Butterfly Curve
Butterfly curve may refer to: *Butterfly curve (algebraic), a curve defined by a trinomial *Butterfly curve (transcendental), a curve based on sine functions {{disambig Mathematics disambiguation pages ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butterfly Curve (algebraic)
In mathematics, the algebraic butterfly curve is a plane algebraic curve of degree six, given by the equation :x^6 + y^6 = x^2. The butterfly curve has a single singularity with delta invariant three, which means it is a curve of genus seven. The only plane curves of genus seven are singular, since seven is not a triangular number, and the minimum degree for such a curve is six. The butterfly curve has branching number and multiplicity two, and hence the singularity link has two components, pictured at right. The area of the algebraic butterfly curve is given by (with gamma function \Gamma) :4 \cdot \int_0^1 (x^2 - x^6)^ dx = \frac \approx 2.804, and its arc length ''s'' by :s \approx 9.017. See also * Butterfly curve (transcendental) The butterfly curve is a transcendental plane curve discovered by Temple H. Fay of University of Southern Mississippi in 1989. __TOC__ Equation The curve is given by the following parametric equations: :x = \sin t \!\left(e^ - 2\cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butterfly Curve (transcendental)
The butterfly curve is a transcendental plane curve discovered by Temple H. Fay of University of Southern Mississippi in 1989. __TOC__ Equation The curve is given by the following parametric equations: :x = \sin t \!\left(e^ - 2\cos 4t - \sin^5\!\Big(\Big)\right) :y = \cos t \!\left(e^ - 2\cos 4t - \sin^5\!\Big(\Big)\right) :0 \le t \le 12\pi or by the following polar equation: :r = e^ - 2\cos 4\theta + \sin^5\left(\frac\right) The term has been added for purely aesthetic reasons, to make the butterfly appear fuller and more pleasing to the eye. Developments In 2006, two mathematicians using Mathematica analyzed the function, and found variants where leaves, flowers or other insects became apparent. See also https://books.google.com/books?id=AsYaCgAAQBAJ&dq=OSCAR+RAMIREZ+POLAR+EQUATION&pg=PA732 * Butterfly curve (algebraic) In mathematics, the algebraic butterfly curve is a plane algebraic curve of degree six, given by the equation :x^6 + y^6 = x^2. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
